Abstract. The Lindstedt series were introduced in the XIX
century in Astronomy to study perturbatively quasi-periodic motions
in Celestial Mechanics. In Mathematical Physics,
after getting the attention of Poincare', who studied
them widely by pursuing to all orders the analysis of
Lindstedt and Newcomb, their use was somehow superseded by other
methods usually referred to as KAM theory.
Only recently, after Eliasson's work, they have been reconsidered
as a tool to prove KAM-type results, in a spirit close
to that of the Renormalization Group in quantum field theory.
Following this new approach we discuss here
the use of the Lindstedt series
in the context of some model problems, like the standard map
and natural generalizations, with particular attention
to the properties of analyticity in the perturbative parameter.